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To: 9.3 Computational speed |
One of the main difficulties in a Lagrangian determination of diffusion processes is that convection can cause a severe deterioration in the spatial distribution of the computational points. For the particle methods, Marshall & Grant [145] and Pelz & Gulak [169] showed that the most important consequence is a loss of accuracy in the quadratures. To maintain accuracy, such methods require a careful monitoring of the particle overlap at all times [119,170]. Solutions for extended times require periodic reinitialization or remeshing of the particle distribution [23,119,170].
The influence of the remeshing process and the time period between remeshings are additional sources of errors and uncertainties. Some particle computations have reported remeshing every six or seven time steps [170]. This suggests the question at which time a Lagrangian computation stops being mesh-free or truly Lagrangian.
The vorticity redistribution method does away with these difficulties. Its only constraint on the point distribution is that a positive solution to the redistribution equations exists. If there is none, a new vortex is added to create one. As a result, the point distribution is implicitly checked at each vortex at each time-step, and restored before it can affect our error estimates.
It is important to reiterate that our computation is truly mesh-free. We simply add a new vortex when we need one. We do not create a new partitioning of the domain; we do not create new quadrature rules based on the new vortex and its neighbors; we do not make any associations between the new vortex and its neighbors.
As discussed in subsection 6.2.4, in the computations presented in section 8.2, we have also searched the existing vortices for any ones that are no longer truly useful, and simply thrown them out, after distributing their vorticity over the neighboring vortices. No other steps were needed.
In our computations, we do not even bother creating a mesh of vortices around our solid bodies. For example, in section 8.3 we compute vortices bouncing off a circular cylinder by merely placing the incoming vortices at the desired initial position. The vortices introduce a slip velocity at the surface of the cylinder, and new vortices are created at the wall to cancel this slip velocity. Our method automatically takes care of extending this vorticity distribution at the wall out into the field. This would work the same way regardless of the number and complexity of the solid bodies present. When the different regions meet, the computational vortices automatically start using the vortices from the other regions.
The mesh-free nature of our computation also allows us to restrict vortices to exactly the regions where we need them. For example, we do not include vortices of zero strength in our computation as particle computations have done [170]. Our impulsively started cylinder computations start without any vortices; the boundary treatment creates the first ones.